Why is my sculpting brush texture/alpha being clipped?
It looks like this is because of two things.
First is the default falloff settings. Just below the texture heading is a falloff heading, shown below:

The default is a smooth falloff, which has a strength of 0 at the edges, even at full hardness I believe. The solution is to switch to a constant falloff. Just be aware that this can cause hard edges on the brush if the texture suddenly gets cutoff...
Which is the second reason. The sculpt brush is round, and your texture is square, meaning that the &quot;corners&quot; of the square will be clipped. The solution is to increase the texture scaling values under the Texture heading. This will scale down the texture a bit so that the square image fits into the round brush.
An alternative is to make the texture itself have a larger black border, so if scaling doesn't work for some reason, that would be the next best thing I think.